Problem: Let $k$ and $m$ be real numbers, and suppose that the roots of the equation \[x^3 - 7x^2 + kx - m = 0\]are three distinct positive integers. Compute $k + m.$
Answer: By Vieta's formulas, the sum of the roots of the equation is $7.$ Furthermore, the only triple of distinct positive integers with a sum of $7$ is $\{1, 2, 4\}.$ To see this, note that the largest possible value for any of the three integers is $7 - 1 - 2 = 4,$ and the only way to choose three of the integers $1, 2, 3, 4$ to sum to $7$ is to choose $1,$ $2,$ and $4.$

Therefore, the roots of the equation must be $1,$ $2,$ and $4.$ It follows by Vieta that \[k = 1 \cdot 2 + 2 \cdot 4 + 1 \cdot 4 = 14\]and \[m = 1 \cdot 2 \cdot 4 = 8,\]so $k+m = 14+8 = \boxed{22}.$